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Journal of Electroanalytical Chemistry

journal homepage: www.elsevier.com/locate/jelechem

Semi-analytical modelling of linear scan voltammetric responses for soluble-insoluble system: The case of metal deposition

Imene Ateka,b, Sunny Mayeb, Hubert H. Giraultb, Abed M. Affounea,⁎, Pekka Peljob,⁎

a Laboratoire d'Analyses Industrielles et Génie des Matériaux, Département de Génie des Procédés, Faculté des Sciences et de la Technologie, Université 8 Mai 1945 Guelma,BP 401, Guelma 24000, Algeriab Laboratoire d'Electrochimie Physique et Analytique, École Polytechnique Fédérale de Lausanne, EPFL Valais Wallis, Rue de l'Industrie 17, Case Postale 440, CH-1951Sion, Switzerland

A R T I C L E I N F O

Keywords:ModellingMetal depositionSoluble-insoluble systemLinear sweep voltammetry

A B S T R A C T

The absence of general theoretical models describing linear sweep voltammetry (LSV) or cyclic voltammetry(CV) responses for soluble-insoluble systems such as one-step electrodeposition reactions under quasi-reversiblecondition makes it difficult to extract quantitative kinetic information from experimental voltammograms. Inthis work, a semi-analytical method for modelling LSV responses for one–step electrodeposition process is de-scribed, for a case where instantaneous nucleation takes place, such as metal deposition on same metal.Voltammetric peaks were analyzed following variation of both dimensionless rate constants and charge transfercoefficients in a broad range. Therefore, kinetic curves for electron transfer processes were established and fittedperfectly by sigmoidal Boltzmann function and linear models. With these models, LSV or CV experimental datacan be used to measure electrodeposition reactions kinetics whatever the degree of reversibility. The Cu(I)/Cu(0)couple in acetonitrile was selected as an experimental example. The model developed in this work predictsaccurately the current response for Cu electrodeposition reaction and an excellent experiment–theory agreementwas found.

1. Introduction

Over the years within the field of electrochemistry, many effortshave been devoted to modelling and simulation techniques to help tounderstand electroanalytical experiments [1–10]. Linear sweep vol-tammetry (LSV) and cyclic voltammetry (CV) are well-known electro-chemical techniques which have played an important role to obtain aclear view about the kinetics, thermodynamics and mechanisms ofelectrode reactions [3–6]. In LSV, the potential is scanned linearlystarting at the initial potential while measuring the current response.CV is an extension of LSV in that the direction of the potential scan isswitched at a predetermined value and the potential is scanned again inthe reverse direction to the initial value. Thus, a triangular potential-time waveform is used in CV [4]. In this framework and according toOldham's earlier work, there are three possible pathways for the es-tablishment of a theoretical voltammograms [5]: analytical modelling,semi-analytical modelling and digital simulation.

For a single-step reaction and particularly for soluble-soluble sys-tems where both oxidized and reduced species are soluble, a com-plementary theoretical studies by either analytical modelling, semi-analytical modelling or by digital simulation, was presented and

extensively discussed for reversible, quasi-reversible and irreversibleprocesses experiments [1–10]. Randles and Ševčík were the pioneers inthis field [11,12]. After that many other investigations have been car-ried out, especially the work of Matsuda and Ayabe for quasi-reversiblesystems [10,13]. A very important theoretical development of cyclicvoltammetry was achieved by Nicholson and Shain in their well-knownpaper published in 1964 [14,15]. For soluble-insoluble system, mod-elling of the voltammograms was first introduced by Berzins and De-lahay [16] who studied the reversible deposition on solid electrodesand derived an analytical expression for current-potential curve forsingle scan, using the Laplace transform and Dawson's integral. Lateron, the theory was extended by Delahay to totally irreversible process[17]. Other researchers have also done similar studies but their mainfocus was for reversible cyclic voltammetry [18–20]. All these devel-opments concerned the redox reactions of simple ions with a singlestep. However, in the case involving amalgam formation, some ap-parent anomalies were noted between experiment and theory for cyclicvoltammetry. These facts were discussed by Beyerlein, and Nicholson[21] and others [22]. Additionally, more complicated models includingion adsorption have been proposed [23].

On the other hand, to describe the transition between fast and slow

https://doi.org/10.1016/j.jelechem.2018.04.021Received 2 March 2018; Received in revised form 9 April 2018; Accepted 12 April 2018

⁎ Corresponding authors.E-mail addresses: affoune.abedmohamed@univ-guelma.dz (A.M. Affoune), pekka.peljo@epfl.ch (P. Peljo).

Journal of Electroanalytical Chemistry 818 (2018) 35–43

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electrode kinetics, the important study related to the main features involtammetric curves such as the peak potential, the peak current andthe half wave potential as a function of normalized heterogeneous rateconstant was done by Matsuda and Ayabe in 1950s. Therefore, threereversibility diagrams were established for soluble-soluble systems[10,13]. In this direction, recently Krulic et al. presented a rough eva-luation of the degree of the reversibility for one electron transfer re-action with an insoluble deposit, considering only the case when theanodic and cathodic charge transfer coefficients are both equal to 0.5[24]. However, the effect of the combination of the rate constant andcharge transfer coefficient have not been reported before this work.

Furthermore, another key aspect for kinetics study was introducedby Nicholson under the so-called working curves theory. This theory isbased on correlation between the peak-to-peak potential in the cyclicvoltammogram and the non-dimensional kinetic rate [15]. Thus, via theuse of the Matsuda and Ayabe diagrams and Nicholson working curves,data quantifying soluble-soluble reaction kinetics could simply be de-termined, whatever the degree of the reversibility. Since no practicaltools are available for soluble-insoluble systems, many authors [25–27]have simply resorted to use the traditional Nicholson approach inconjunction with Matsuda and Ayabe criteria to analysis experimentalvoltammograms under quasi-reversible condition. This approach,however, leads to misinterpretation of the experimental data.

For these reasons and to remedy these shortfalls, the purposes of thepresent paper are:

- Modelling of LSV responses for quasi-reversible soluble-insolublesystem, employing a semi analytical method.

- Developing a general model to extract the kinetic parameters viaLSV or CV data, whatever the degree of reversibility.

However, LSV models proposed in this work are only applicable to asituation where instantaneous nucleation takes place, such as metaldeposition on same metal, or for example silver deposition on gold insome specific conditions [28]. If nucleation overpotential is required toinduce the nucleation, followed by for example progressive 3D nu-cleation, the shape of the voltammogram will change drastically [24].

Finally, to demonstrate the accuracy and the effectiveness of ournumerical model, experimental tests with the system Cu(I)/Cu(0) areperformed. The accurate estimation of the electrochemical kinetics ofthis process is required to understand better the limitations for the non-aqueous batteries utilizing Cu(I)/Cu(0) couple as the negative redoxcouple [29,30].

A further complication not considered in this works is the aptlynamed “enigma of metal deposition” [31], namely, the fact that thesolvation environment of the ion changes completely upon metal de-position, while in electron transfer reactions of soluble-soluble systemsonly slight solvent reorganization is required. Despite this difference,some metal deposition reactions are among the fastest electrochemicalreactions [31]. This problem has been described in detail by Gileadi[31–34], and has been studied both experimentally and theoreticallyfor amalgam formation reactions, including copper deposition fromvarious nitriles by Fawcett et al. [35,36]. To conclude, simplifiedButler-Volmer type formalism employed in this work yields apparentrate constants, while more detailed analysis including corrections fordouble layer effects would be required to obtain the standard rateconstant [35,36]. A solution for the “enigma of metal deposition” wasrecently proposed by Schmickler et al. based on theoretical calcula-tions, proposing that univalent metal cations can approach very close tothe metal surface without loss of solvation energy [37]. At this positionthe ion is close enough to experience a strong energetically favourableelectronic interaction with the identical atoms of the electrode surface,allowing overcoming the large energy required to remove the solvationshell [37].

2. Theory

Prior to solving and calculating LSVs for soluble-insoluble system,we give a brief overview of Nicholson-Shain model currently used forextracting of detailed information about kinetics and mass transport forelectrode reactions where both oxidized (O) and reduced (R) species aresoluble in the solution. The reversibility factor is defined as [10]:

=( )

k

D vΛ

π nT

0

OF

R

1/2

(1)

For reversible reactions (Λ > 15)

= ⎛⎝

⎞⎠

∗I n AC nT

D v0.4463 F FRp O

1/2

O1/2 1/2

(2)

For irreversible reactions (Λ≤ 10−2(1+α), i. e. Λ≤ 0.015ifα=0.5)

= ⎛⎝

⎞⎠

∗I n AC αnT

D v0.4958 F FR

αp O

1/2

O1/2 1/2

(3)

Throughout this paper, we seek to apply Nicholson's method togenerate LSV model for reactions involving an insoluble species.Nicholson's method, also called semi-analytical method, generallyconsists of solving the mass transport problem analytically, and de-riving the theoretical voltammetric relationships numerically [14,15].In the present work, our attention focused on single step electro-deposition reaction. Thus, a soluble-insoluble system of metallic ionsMn+ and metal M couple could be represented as Mn+(sol)+ ne−⇄M(s).

Upon neglecting migration and convection contribution, the pro-blem of the mass transport of metallic cations in Cartesian coordinatescan be described by the partial differential equations (PDE):

∂∂

=∂

∂+

++C x t

tD

C x tx

( , ) ( , )MM

2M

2

nn

n

(4)

With the following initial and boundary conditions:

= = ∗+ +t C x C0, ( , 0)M Mn n (5)

→ ∞ ∞ = ∗+ +x C t C, ( , )M Mn n (6)

→ = − ⎡⎣⎢

∂∂

⎤⎦⎥ =

++

x I tn A

DC x t

x0, ( )

F( , )

xM

M

0

nn

(7)

The current change is given by Butler-Volmer equation:

= ⎡⎣⎢

⎛⎝

− − ⎞⎠

− ⎛⎝

− − ⎞⎠

⎤⎦⎥

+

I t nFAk C t α nFRT

E t E

C t αnFRT

E t E

( ) (0, ) exp (1 ) ( ( ) )

(0, ) exp ( ( ) )

0M

0

M0n

(8)

The electrode potential E(t), is swept linearly with scan rate v in thenegative direction starting at initial potential Ei which no electrodereactions occur, so that the potential at any time t is given by:

= −E t E vt( ) i (9)

Let us consider that at the initial state, the equilibrium is achieved atthe surface of electrode, so the equilibrium potential is given by theNernst equation:

⎜ ⎟= = + ⎛⎝

⎞⎠

+E E E RT

nFaa

lni eq0 M

M

n

(10)

where aM=1 for pure metal and =+ ++a γ

C

CM Mn nnM0 are respectively the

activity of the metal and the corresponding cation, γi is the activitycoefficient of the species i and C0 is the standard concentration of1mol L–1. The concentration of metal CM in Eq. (8) can be expressed asCM= aMC0= C0 assuming that the activity coefficient of the metal is 1.

In order to solve the above equations system (Eqs. (4)–(10)), wesummarize briefly our solution procedure in three steps as follows:

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Step 1: converting Fick's differential law into Integral Equations.Before the computation of the current-potential curve, it is neces-

sary to know the concentration as a function of time. After applyingLaplace's transform to PDE Eq. (4) under the initial condition (5) andthe boundary conditions (6)–(7), the solution has the form:

∫= +−

∗+ ++

C t CnFA πD

I τt τ

dτ(0, ) 1 ( )t

M MM 0

n nn (11)

and remembering that CM(0, t)= C0.Step 2: converting the linear sweep voltammetry parameters into a

non-dimensional form.For modelling purposes, it is helpful to convert the dimensioned

variables into a non-dimensional form. Thus, we set:Dimensionless initial potential, Init:

= −Init nFRT

E E( )i0

(12)

Dimensionless applied potential, Φ:

= − = −Φ nFRT

E t E Init σt( ( ) )0(13)

Dimensionless scan rate, σ:

=σ nFRT

v (14)

Step 3: calculation of the current.From combination of Eqs. (1)–(14), we obtain the general equation

for the current:

= ⎛⎝

⎞⎠

∗ + +I t nFAC πD nFvRT

σt( ) ( ) Ψ( )M M1/2

1/2n n

(15)

where the dimensionless current Ψ(σt)

=∗ + + ( )

σt I t

nFAC πDΨ( ) ( )

( ) nFvRTM M

1/2 1/2n n (16)

is given by the following integral:

∫ −= −⎡

⎣− − + ⎤

⎦z

σt zdz σt

ωS σt S σtΨ( ) 1 Ψ( ) 1 [ ( )] [ ( )]

σtα

0 (17)

With the dimensionless heterogeneous rate constant defined as de-scribed in by Krulic et al. [24]:

=( )

ω k

θ πD vα nFRT

0

1/2

(18)

where θ is expressed as:

= ⎛⎝

⎞⎠

− = ∗ +θ nFRT

E E C Cexp [ ] /i0

M0n

(19)

and

= −S σt σt( ) exp( ). (20)

In this work the initial potential Ei is used to calculate ∗ +C Mn asactivity of the metal is always taken as 1. For performing the calculationof the integrals in the expression (17), the same numerical methoddeveloped by Nicholson and Shain [14] was used which leads to thefollowing LSV algorithm:

∑+ − + −

= − − − +

= …

=

−

+

Ψ K K j Ψ j Ψ j

δΨ δK ω S δK S δK

k N

(1) [ ( 1) ( )]

12

[ 1 ( ) [ ( )] [ ( )] ],

1, 2, 3

j

K

α α β

1

1

(21)

Thus, this algorithm enables the dimensionless voltammograms tobe calculated, provided that the values of Init, Φ, ω and α are known.

3. Material and methods

3.1. Chemicals

All solvents and chemicals were used as received without furtherpurification. The solvent was acetonitrile (ACN, extra dry over mole-cular sieves, 99.9%, from Acros). Electrodes were made with copperwire (dia. 1 mm, ≥99.99%, from GoodFellow). The supporting elec-trolyte was tetraethylammonuim tetrafluoroborate (TEABF4, 99%, fromABCR) and the electroactive species was tetrakis(acetonitrile)copper(I)tetrafluoroborate ([Cu(CH3CN)4]BF4,> 98%, from TCI).

3.2. Electrochemical measurements

LSV measurements were performed by using one-compartmentthree-electrode cell where the reference electrode and auxiliary elec-trode were coils of 1 mm diameter copper wire and with a copper discserving as a working electrode. The working electrode was prepared byheat-sealing 1mm diameter copper wire in 2mm diameter glass tube,followed by polishing with consecutively finer abrasive papers followedby different sizes of alumina, down to 0.05 μm particles, on polishingcloths. The electrolyte consisted of a mixture of given concentrations of[Cu(CH3CN)4]BF4 as a copper source, 100mM TEABF4 as a supportingelectrolyte, and acetonitrile as a solvent. All potentials are expressed vs.the Cu wire in the solution of the given concentration of [Cu(CH3CN)4]BF4. Hence, the potential of the Cu reference in equilibrium with 10mMCu+ solution in acetonitrile on the “non-aqueous standard copperelectrode” scale (SCuE), i.e. vs. Cu+ solution with the activity of 1 inequilibrium with Cu in acetonitrile is assumed to follow the Nernstequation considering the γCu+ = 1, i.e. the potential of the reference is−0.118 V vs. SCuE in acetonitrile. To establish the relation with theferrocene (Fc) scale recommended by IUPAC, the potential of the Fc+/Fc couple vs. Cu/10mM Cu+ in ACN was measured as 0.69 ± 0.01 V,i.e. the potential of our reference electrode is −0.69 V vs. Fc+/Fc. LSVdata for the reduction of copper (I) in acetonitrile were recorded withan Autolab model PGSTAT302N potentiostat equipped with theSCAN250 analog scan generator module. We point out that, all ex-perimental steps (solution preparation, LSV measurement) were per-formed at ambient temperature (25 °C) under anaerobic conditionsusing a nitrogen filled glove box, utilizing positive feedback iR com-pensation.

4. Results and discussion

4.1. Theoretical results

4.1.1. Effect of the kinetic rateGenerally, the critical parameters used for the diagnosis of electron

transfer reactions via linear sweep voltammograms are the magnitudesof the peak current, the peak potential and the half peak width. TheseLSV responses may depend on multiple variables including the chargetransfer coefficient α, the potential sweep rate v, and the heterogeneousstandard rate constant k0. In the present paper, the mutual influence ofk0 and v is expressed through the magnitude of the dimensionlessparameter, ω, defined by Eq. (18). The effects of varying ω for a con-stant value of α on the position, high and width of peak are examinedand shown in Fig. 1, in which three distinct regions can be seen:

1) ω≥ 103

For which, the dimensionless current-potential curves become in-sensitive to the dimensionless rate constant (curves a, b, c in the inset ofFig. 1). The dimensionless peak current, π1/2Ψp, takes a constant valueof −0.6105, as reported by Berzin and Delahay [16] for reversiblesystem.

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2) ω≤ 10−3

In this region, the calculated linear sweep voltammograms retaintheir shape (curves i-k in Fig. 1). The peak half width, ΔΦp/2, remainsunchanged and the dimensionless peak current (π1/2Ψp) keeps stable invalue of −0.350, which is in good agreement with literature values forirreversible systems [17]. Contrary to the peak shape, the peak positionis shifted towards more negative potentials as ω decreases.

3) 10−3 < ω < 103

Within these two kinetic regions (curves d–h in Fig. 1), a decrease inω leads to a diminution in the peak height, an increase in peak halfwidth, and a shift of the dimensionless peak potential (Φp) towardsmore negative values. These observations refer to characteristics ofquasi-reversible waves.

4.1.2. Effect of the charge transfer coefficientA series of theoretical voltammograms in which the charge transfer

coefficient α is varied, are displayed in Fig. 2. Note that the impact of αon the peak parameters depends upon the magnitude of the di-mensionless kinetic rate ω.

- For ω=103, no change occurs in current and potential values withα varying from 0.2 to 0.8. This observation agrees reasonably withprevious studies for reversible system [16].

- For ω=1, an increase in α value from 0.2 to 0.8 is followed by aslightly increase in the dimensionless peak current (in absolutevalue), while the peak potential remains almost constant.

- For ω=10−3, the effect of the electron transfer coefficient has anappreciable influence on the height, position, and the shape of thepeak.

4.1.3. Kinetics curves: coupling effects of kinetic rate and charge transfercoefficient

It is clear from the above results that the voltammograms in theintermediate region, 10−3 < ω < 103, are qualitatively different.Furthermore, until now, as no characteristic equations or practical toolsfor analysis of experimental voltammograms for quasi-reversible sys-tems has been reported, our goal in this section is to provide a general,simple and direct model capable to solve the problem of the determi-nation of the kinetic parameters in the case of quasi-reversible soluble-insoluble systems. The same issue was encountered before for soluble-soluble system, and the solution was offered by suggesting series ofworking curves which covered the range from reversible to irreversible

Fig. 1. Calculated linear sweep voltammograms for various value of ω with α=0.5; a: ω=105, b: ω=104, c: ω=103, d: ω=102, e: ω=101, f: ω=1, g:ω=10−1, h: ω=10−2, i: ω=10−3, j: ω=10−4, k: ω=10−5.

Fig. 2. The effect of transfer coefficient α on theoretical voltammograms.

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behaviour.Likewise, in the next set of figures, we have constructed three ki-

netic curves for single electron transfer process, in which we showquantitatively the coupling effects of the dimensionless kinetic rate (ω)and the electron transfer coefficient α on LSV responses. The values of ωin these calculations was varied from 106 to 10−6 while the values of αwas varied from 0.2 to 0.8. Fig. 3 shows the plot of the peak currentratio Ψp/(Ψp)rev vs. log(ω), where (Ψp)rev is the reversible dimension-less peak current. Fig. 4 shows quantitatively the variation of the peakshape through the half peak width changes, = −Φ E EΔ ( )n

Tp/2F

R p p/2 as afunction of log(ω) and α and Fig. 5 describes how the cathodic peakposition ηp, = −η E E( )n

T p eqpF

R , changes as a function of log(ω) and α.It is apparent from Fig. 4, that the plots of the LSV responses as a

function of both, the dimensionless heterogeneous rate constant ω andthe charge transfer coefficient α, exhibit sigmoidal shapes. Data werethen fitted perfectly by the sigmoidal Boltzmann functions:

= + −

+ ⎡⎣ ⎤⎦− − −

αΨ(Ψ )

1 (0.811 1)

1 exp X αα

p

p rev

0.5

( 0.528 0.099)0.477 0.248 (22)

− = + −

+ ⎡⎣

⎤⎦

−

+ −

αΔΦ 0.770 (1.857 0.770)

1 exp X αα

p/21

(0.557 )0.445

0.2160.316 (23)

where: X= log (ω). Each of these fit models enables voltammetricquantification of the electrode kinetics from simple peak current andpeak potential measurements of the experimental linear sweep vol-tammograms for either reversible, quasi reversible or totally irrever-sible electron transfer process, provided that the value of α is known orcan be estimated accurately. Also, from Figs. 3–5, the following kineticzones properties can be concluded:

In the zone A both the current and the peak potentials are in-dependent of the values of α and the dimensionless kinetic rate ω. Allpeak parameters ηp, Ψp, and ΔΦp/2, reach their reversible values,yielding to the following reversible criteria:

For ω≥ 103:

= ⎛⎝

⎞⎠

∗ + +I nFAC D nFvRT

0.6105 ( )p M M1/2

1/2n n

(24)

= −E E RTnF

0.854p eq (25)

− = −E E RTnF

0.77p p/2 (26)

These criteria for kinetically reversible system are in excellentagreement with those of previous theoretical investigations[16,18–20,38].

Quasi-reversible features are observed in the zone B (−3 < logω < 3), where marked changes in the various peak parameters as afunction of both log ω and α are evident. As shown in Figs. 3–4, thepeak height ratio increases while the half peak width decreases forincreasing values of α. Both values decrease with decreasing values oflog ω. This effect of the charge transfer coefficient can be interpreted bythe change in the symmetry of the energy barrier.

Irreversible behaviour is evident in the zone C, where the peakcurrent and half peak width remain constant with decreasing log ω forspecific values of α while the peak potential continues to decrease lin-early as the function of decreasing log ω. As for the quasi-reversiblecase, the peak height ratio and peak potential increase and the half peakwidth decreases for increasing values of α. Therefore, the followingconclusions can be drawn:

For ω≤ 10−3:

=Ψ

Ψα

( )0.811p

p rev

1/2

(27)

Fig. 3. Variation of the peak current ratio, Ψp/(Ψp)rev, as the function of thedimensionless rate constant for several values of α. Solid lines are best fits to thesigmoidal Boltzmann functions with a correlation coefficient of 0.99. Zones A,B, C denote the reversible, quasi-reversible and totally irreversible zones, re-spectively.

Fig. 4. Dependence of the half peak width of linear sweep voltammograms(ΔΦp/2) on the logarithm of the kinetic parameter ω for various α values. Solidlines are best fits to the sigmoidal Boltzmann functions with a correlationcoefficient of 0.98.

Fig. 5. Plots of the reduction peak position (ηp) against log(ω) for various αvalues.

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= −Φ αΔ 1.857p/21 (28)

These equations show a very good agreement with those obtainedby Delahay for irreversible soluble–insoluble redox system [17].

A further key point to consider in Fig. 5, which was also observedand demonstrated by Krulic et al. [24], is that in the region when logω≤−1), the magnitude of ηp depends linearly on the logarithm of thekinetic parameter ω. Therefore, by using linear regression approxima-tions, the dependency of ηp on ω and α could be expressed as follows:

= − +αη X α2.303 [0.115 log( ) 0.78]p (29)

This equation coincides with that established by Krulic for α=0.5[24], it can be used for estimation of the kinetic rate constant, in par-ticularly for quasi-reversible and irreversible electrodeposition pro-cesses if the value of α is known or can be estimated.

4.2. Experiment–theory comparison

In order to test the validity of our numerical approach to extractkinetic and mass transport parameters from LSV data, an example of Cuelectrodeposition reaction in organic solution is presented. The Cusystem includes a simple one-electron transfer reaction according to[29]:

+ →+ −Cu e Cu(0) (i)

although, the removal of the complexing solvent should also be con-sidered [35,36]. Fig. 6a depicts typical LSV profiles collected at dif-ferent scan rates for reduction reaction, Cu(I)/Cu(0), in acetonitrile.Regarding to the peak heights, an increasing trend was observed onincreasing the scan rates. Furthermore, the peak potential was seen toshift gradually towards more negative potential values over25–200mV/s scan rate suggesting a quasi-reversible character. To de-duce the mass-transport and kinetic proprieties of the Cu(I)/Cu(0)system, the unknown parameters values: DCu(I), α and k0 need to bederived. There are two ways for the calculation of DCu(I), α and k0: byeither computationally or by adjusting curves based only on LSV al-gorithm (Eq. (21)) and different combinations of DCu(I), α and k0.

4.2.1. Utilization of working curvesFirst, we used the common employed procedure, the semi-in-

tegrative voltammetry, for the calculation of Cu(I) diffusion coefficientand a direct Tafel analysis for the measurement of the transfer coeffi-cient α. Fig. 6b shows a sigmoidal curve (dotted line) obtained fromsemi-integration of typical voltammetric current recorded at 100mV/s.It should be noted that we have employed the Saila methodology [39]for establishing semi-integrated plots. As shown in Fig. 6b and as perprinciple of semi-integration technique, the semi-integration of thevoltammetric current responses with respect to time yields to the sig-moidal-type curve with a plateau. This plateau level represents thelimiting semi-integral current, of height m∗:

Fig. 6. (a) LSV curves of electrodeposition of Cu(I) on Cu disc electrode from 10.25 mM of tetrakis(acetonitrile)copper(I) tetrafluoroborate in acetonitrile at variousscan rates. (b) LSV curve of reduction reaction of Cu(I) recorded scan rate of 100mV/s. The dotted line indicates semi-integrated current for forward scan. (c) Tafelcurve derived from the rising part of the LSV curve at scan rate of 100mV/s. The inset shows the LSV data used for drawing the Tafel curve. The potential is expressedvs. the Cu reference in equilibrium with 10.25mM Cu+ in solution, i.e. −0.118 V vs. standard copper electrode in acetonitrile or −0.69 V vs. Fc+/Fc.

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=∗ ∗+ +m nFAC DM M

1/2n n (30)

and the diffusion coefficient of Cu(I) in acetonitrile at 25 °C was com-puted utilizing this equation (as shown in Table 1) by considering thefollowing experimental values: n=1, CCu+

∗=10.25 mM. Fig. 6c de-picts the representation of the Tafel plot obtained from experimentalLSV data, at a scan rate of 100mV/s. Using the slope calculated fromFig. 6c and the Tafel equation, the value of α was calculated and theresult is reported in Table 1. In the next step, a more detailed ex-amination of the voltammetric peaks allowed us to quantify the stan-dard rate constant (k0 in cm2/s) for the Cu(I)/Cu(0) redox couple ac-cording to the models presented in Section 4.1.3. The results are listedin Table 1. With these values, Eqs. (22), (23) and (29) were used toobtain the final value for the k0.

4.2.2. Fit and simulationThe parameters, DCu(I), α and k0 were also obtained by comparison

of theoretical and the experimental linear scan voltammograms (in100mM TEABF4 and 10mM Cu(CH3CN)4BF4), using the LSV modelalgorithm with a series of adjustments in the input values. For thispurpose, theoretical voltammograms were obtained by utilizing thediffusion coefficient of Cu ions within the range reported in the lit-erature [40], and varying the values for k0 and α for the Cu(I)/Cu(0)system as these values have not been reported previously in the lit-erature. In our work, α was varied from 0.1 to 0.9 while the di-mensionless kinetic rate was varied from 10−3 to 103.

In the purpose to compare the two approaches: i) theoretical fittingand ii) working curves, and to illustrate the LSVs are obtained with theparameters from Eqs. (22), (23) and (29), an analysis study was per-formed first with the voltammetric data collected at 100mV/s. Toachieve best fit to experimental LSV data obtained at 100mV/s, wetested various values of DCu(I), α and k0. With the following parameters:DCu(I) = 1.7×10−9m2s−1, α=0.8 and k0= 5.12× 10−5 cm s−1

, theresultant theoretical voltammogram is in very good agreement with theexperimental LSV curve (see Fig. 7a). To demonstrate the applicabilityof the working curves, theoretical voltammograms were calculated withEq. (21) with the parameters shown in Table 1 and the results arepresented in Fig. 7b. Better agreement was obtained by using Eqs. (22)and (23) than by Eq. (29). The small discrepancy in kinetic predictionusing Eq. (29) could be attributed to the dimensionless kinetic rateconstant ω of copper reduction which was just out of range of

applicability of Eq. (29). We have found above that Eq. (29) is valid forω≤ 0.1; however the ω value calculated for Cu reaction is 0.15.

Table 1 shows the obtained parameters for the reduction of Cu(I) toCu(0) in acetonitrile at 100mVs−1. The obtained DCu(I) values are si-milar with the values previously reported in literature [40], althoughslightly higher diffusion coefficient was obtained from semi-integrationanalysis. Furthermore, the calculated values of k0 indicated that the Cudeposition process in acetonitrile is quasi-reversible.

Likewise, the kinetics of the Cu(I)/Cu(0) system were analyzed at allscan rates. The use of Eqs. (22), (23) and (29), yielding average valuesof the standard rate constant k0: 4.86(± 0.68)× 10−5,4.63(± 0.22)× 10−5 and 7.76(± 0.76)× 10−5 cm/s, respectively.Excellent fits were reproduced between the experimental LSVs and si-mulated ones under various scan rates (see Fig. 8) and the averagefitted k0 value was found: 5.43(± 0.52)× 10−5 cm/s. These results

Table 1Kinetic-mass transport parameters for Cu(I)/Cu(0) redox couple.

Cu(I)/Cu(0) system DCu(I)[10−9m2s−1] α k0[10−5 cm s−1]SI Fit Tafel Fit Working curves Fit

Eq. (22) Eq. (23) Eq. (29)1.75 1.70 0.82 0.80 3.84 4.99 7.73 5.12

Fig. 7. (a) A comparisons between theoreticallinear scan voltammogram and experimentalvoltammogram obtained and recorded at100mV/ s scan rate. The parameters used fortheoretical prediction are: DCu(I) =1.70× 10−9 m2s−1, α=0.8 andk0= 5.12× 10−5 cm s−1. (b) Comparisonof theoretical LSV responses with experimentaldata using Fit models presented by Eqs. (22),(23) and (29). Parameters used for LSV model-ling: DCu(I)=1.75×10−9 m2s−1α=0.82 andk0=3.83×10−5 cm s−1 calculated from Eq.(22), k0=3.83×10−5 cm s−1 calculatedfrom Eq. (23) and k0=5.88×10−5 cm s−1

calculated from Eq. (29). The potential is ex-pressed vs. the Cu reference in equilibrium with10.25mM Cu+ in solution, i.e. −0.118V vs.standard copper electrode in acetonitrile.

Fig. 8. Fits obtained between experimental LSV data (solid line) and theoreticalLSV data (dashed line) generating using Eq. (21), for parameters (α and DCu(I))shown in Table 1, under different scan rates.

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confirm the above conclusions, and reveal that the working curvesprovided in this work can be used to accurately analyze the linear scanvoltammetric current responses for quasi-reversible soluble/insolublesystem.

The high transfer coefficient value of 0.8 obtained in this work is inline of the values obtained for Cu deposition on amalgams from nitrilesolvents, indicating that the reaction site is located in the inner part ofthe electric double layer [35,36]. However, the apparent rate constantsmeasured in this work for Cu deposition are three orders of magnitudesmaller than the standard rate constants obtained for copper depositionon mercury [35,36], indicating that ion transfer into mercury is easierthan nucleation on copper surface.

5. Conclusion

In summary, using an extension of Nicholson's method, a new LSValgorithm has been proposed for use in the computing of voltammo-grams in the case of soluble-insoluble system where the electrontransfer takes place in a single step. The effect of the kinetic parameterson the LSV responses has been examined. Through the variation of thepeak parameters with dimensionless kinetic rate (ω) and the transfercoefficient (α), series of kinetic curves have been established. The ob-tained results showed that according to the magnitude of the di-mensionless rate constant the various LSV responses limitation could bedivided into three zones:

Zone A: ω≥ 103, reversible process.Zone B: 10−3 < ω < 103, quasi-reversible process.Zone C: ω≤ 10−3, irreversible process.Moreover, we offer here three working curves to extract kinetic

details with high accuracy and simplicity to use, as only the experi-mental values corresponding to the peak high, peak position and peakwidth are required.

LSV models proposed in this work are applicable only to a situationwhere instantaneous nucleation takes place, such as metal depositionon same metal, or for example silver deposition on gold in some specificconditions [28]. If nucleation overpotential is required to induce thenucleation, followed by for example progressive 3D nucleation, theshape of the voltammogram will change drastically, and the describedkinetic curves and LSV models are no longer valid [24]. On the otherhand, we point out that although this paper was limited to investigationof the soluble-insoluble system during a linear scan, the LSV algorithmcould be extend with some modifications to model also CV, but in thiscase the activity of the metal as the function of surface coverage wouldbe required.

Acknowledgements

I. A. is grateful for the financial support from Ministère del'Enseignement Supérieur et de la Recherche Scientifique, RépubliqueAlgérienne Démocratique et Populaire, Programme NationalExceptionnel, Algeria. P.P. and S.M. acknowledge the financial supportfrom the Swiss National Science Foundation under Grant AmbizioneEnergy 160553.

Appendix 1. Nomenclature

Notations

CM Concentration of the metal MCMn+ Concentration of the metallic ions Mn+

∗ +C Mn Bulk concentration of the metallic ions Mn+

C0 Standard concentrationDMn+ Diffusion coefficient of metallic ions Mn+

A Electroactive surface arean Number of electronsF Faraday's constant

R Universal gas constantT Absolute temperaturek0 Standard rate constantI CurrentE(t) Electrode potentialEi Initial potentialE0 Standard potentialEeq Equilibrium potentialv Potential scan ratet Time

Greek letters

α Charge transfer coefficient1–α Anodic charge transfer coefficientω Dimensionless kinetic rate parameterΨ Dimensionless currentη Dimensionless overvoltage (η= nF/RT(E− Eeq))σ Dimensionless scan rateγMn+ Activity coefficient of the metallic ions Mn+

δk Small time interval

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